Most phones will happily take up to 2A in power. Your Arduino can supply around 200 mA without overheating, so for proper charging you'll have to use an external power supply connecting to the phone, not your Arduino. At 200 mA it's also going to take a really long time before your phone is charged.
By the way, new smart phones can be had for well under USD 200 if not USD 150 by now... sounds to me like the cheaper option the moment you take your time into account developing this charger.